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Worksheet: Fundamental Theorem of Line Integrals

Q1:

Is there a potential 𝐹 ( π‘₯ , 𝑦 ) for f i j ( π‘₯ , 𝑦 ) = 𝑦 βˆ’ π‘₯ ? If so, find one.

  • A yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 + 𝐾
  • B yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦
  • C yes, 𝐹 ( π‘₯ , 𝑦 ) = 𝑦 2 βˆ’ π‘₯ 2 2 2
  • Dno
  • E yes, 𝐹 ( π‘₯ , 𝑦 ) = 𝑦 2 βˆ’ π‘₯ 2 + 𝐾 2 2

Q2:

State whether or not the vector field f i j k ( π‘₯ , 𝑦 , 𝑧 ) = π‘Ž + 𝑏 + 𝑐 where π‘Ž , 𝑏 , 𝑐 are constants has a potential in ℝ 3 .

  • Ayes
  • Bno

Q3:

Is there a potential 𝐹 ( π‘₯ , 𝑦 ) for f i j ( π‘₯ , 𝑦 ) = ( 8 π‘₯ 𝑦 + 3 ) + 4 ο€Ή π‘₯ + 𝑦  2 ? If so, find one.

  • A yes, 𝐹 ( π‘₯ , 𝑦 ) = 4 π‘₯ 𝑦 + 1 2 𝑦 + 3 π‘₯ 2 2
  • B no
  • C yes, 𝐹 ( π‘₯ , 𝑦 ) = 4 π‘₯ 𝑦 βˆ’ 1 2 𝑦 + 3 π‘₯ 2 2
  • D yes, 𝐹 ( π‘₯ , 𝑦 ) = 4 π‘₯ 𝑦 + 2 𝑦 + 3 π‘₯ 2 2
  • E yes, 𝐹 ( π‘₯ , 𝑦 ) = 4 π‘₯ 𝑦 βˆ’ 2 𝑦 + 3 π‘₯ 2 2

Q4:

Is there a potential 𝐹 ( π‘₯ , 𝑦 ) for 𝑓 ( π‘₯ , 𝑦 ) = ο€Ή π‘₯ π‘₯ 𝑦 + 2 π‘₯ π‘₯ 𝑦  + π‘₯ 𝑦 3 2 c o s s i n i j ? If so, find one.

  • A no
  • B yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ π‘₯ 𝑦 βˆ’ 2 π‘₯ 𝑦 2 s i n c o s

Q5:

Is there a potential 𝐹 ( π‘₯ , 𝑦 ) for f i j ( π‘₯ , 𝑦 ) = ο€Ή 𝑦 + 3 π‘₯  + 2 π‘₯ 𝑦 2 2 ? If so, find one.

  • A yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 + π‘₯ 3 2
  • B no
  • C yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 βˆ’ π‘₯ 3 2
  • D yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 + π‘₯ 2 3
  • E yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 + 𝑦 π‘₯ 3 3

Q6:

Is there a potential 𝐹 ( π‘₯ , 𝑦 ) for f i j ( π‘₯ , 𝑦 ) = π‘₯ βˆ’ 𝑦 ? If so, find one.

  • A yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ 2 + 𝑦 2 2 2
  • B no
  • C yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ βˆ’ 𝑦 2 2
  • D yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ 2 βˆ’ 𝑦 2 2 2
  • E yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ + 𝑦 2 2

Q7:

Which of the following is true concerning the vector field V ∢ ℝ β†’ ℝ   defined by V i k ( π‘₯ , 𝑦 , 𝑧 ) = 𝑦 βˆ’ π‘₯ ?

  • A V has a positive divergence at every point ( π‘₯ , 𝑦 , 𝑧 ) ∈ ℝ  .
  • B V ( π‘₯ , 𝑦 , 𝑧 ) β‰  0 , for all ( π‘₯ , 𝑦 , 𝑧 ) ∈ ℝ  .
  • C V is a conservative vector field.
  • D V ( π‘₯ , 𝑦 , 𝑧 ) is orthogonal to the vector j ⟨ 0 , 1 , 0 ⟩ , for all ( π‘₯ , 𝑦 , 𝑧 ) ∈ ℝ  .

Q8:

Is there a potential 𝐹 ( π‘₯ , 𝑦 ) for f i j ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 βˆ’ π‘₯ 𝑦 2 3 ? If so, find one.

  • A yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 2 βˆ’ π‘₯ 𝑦 2 2 2 3 2
  • B yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 2 + π‘₯ 𝑦 2 2 2 3 2
  • C yes, 𝐹 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 2 2
  • D no
  • E yes, 𝐹 ( π‘₯ , 𝑦 ) = βˆ’ π‘₯ 𝑦 2 2

Q9:

State whether or not the vector field f i j k ( π‘₯ , 𝑦 , 𝑧 ) = π‘₯ 𝑦 βˆ’ ο€Ή π‘₯ βˆ’ 𝑦 𝑧  + 𝑦 𝑧 2 2 has a potential in ℝ 3 .

  • Ano
  • Byes